When I was in late primary I was introduced to squares and square roots, at first this was a case of “if you have a number and want to square it, multiply the number by itself”. Being competent at multiplication, this was straightforward. Next was square roots, given as “what would you have squared to get the given number”, a little trickier but again like most I found this fine because I was already competent at multiplication.
My understanding of roots at this stage was purely operational, √5 to me was not a “number” but rather an instruction “find the number you started with and squared to get 5″. At this stage I don’t think there is anything wrong with that, but at some point(s), my relationship with √5 shifted to that of a position on the number line and this gave the ability to manipulate √5 in surd form.
It is vitally important that students are able to calculate roots (skills are important), but we should also be looking for opportunities to shift the relationship that students have with mathematical objects.
This is why students need a maths teacher! And this is part of the job of a maths teacher, this is where expertise of mathematics is important. Teachers must plan for this and take opportunities when presented, by planning tasks that offer more of these opportunities we are providing students with the opportunity to shift their relationship with mathematical objects and ideas.
Here are some tasks that I believe provide opportunities to shift mathematical thinking.
One thing I am hoping for here is students moving towards an understanding of the composition of a number from factors. Another is using combinatorics and the prime factorisation.
Questions I ask would depend on the students and also what I want to get out of the task. I may be happy for some students to draw the boxes to check number of squares, for some I might pose the question “how many will 2^10 or 3^a x 5^b have?”
In this task I hope for students to develop an understanding of place value, but there might also be opportunity to develop an understanding of commutativity, and also linking back to prime factorisation.
The idea that an infinite sum has a finite limit was not obvious to me at school, the diagram aids this. I may choose to start this task with just finite sums, eg start with 1/2 + 1/4 using the diagram, then 1/2 + 1/4 + 1/8, this would have the benefit of encouraging formal addition of fractions using the diagram as an aid.
Here are a few things I became aware of as a child;
I can double one number and half the other to get the same answer when doing multiplication, this later became ab = (a/c)(bc).
This “trick” doesn’t work with addition, I don’t think this is obvious to students.
I can add any number to one number and take the same away from the other and get the same answer when doing a sum, this later became a + b = (a + c) + (b – c)
This manipulation with numbers made it clear to me why the “rules” of arithmetic with fractions and then algebraic expressions have to be so. I think this task could aid this development as well as be used to introduce completing the square (thanks to Colin Hegarty for that idea).
With this task I would be encouraging students to explore similar triangles and ratio. I may give / suggest some pythagorean triples to ease this investigation. I may draw (to scale to check or not) an enlargement of a student’s rectangle and ask them to predict different lengths. Another question I might ask is “can you draw a rectangle / right angled triangle to represent ….” then give some large field or something.
A lot of students will notice and be able to justify this, with this task I usually try to introduce a more formal method of proof. I remember having a bit of an epiphany myself when being taught proof by induction that all natural number could be thought of as the previous number, plus one. It still fascinates me today that this can’t be done with the reals.
Hi Sam
Apologies fir missing this blog of yours – I find it very interesting and feel it is a great basis for an MT article.
If you had any work from any of your learners, to show what they typically did with any of these tasks that would be a bonus, but not necessary – the key issue is to get the kind of message that JM was talking about on Saturday at Leicester out to a wider audience – specifically his quote from 2005: “……a lesson without the opportunity for learners to express a generality is not in fact a mathematics lesson.” Mason et al (2005, 297), Developing thinking in algebra (Open University Press)
There is something I am not clear about and this is the statement: Questions I ask would depend on the students and also what I want to get out of the task. I may be happy for some students to draw the boxes to check number of squares, for some I might pose the question “how many will 2^10 or 3^a x 5^b have?”
Anyway, great to spend time with you on Frid and Sat. Cheers MIke
Hi Sam
I like the bit about root(5) being an instruction, and then not, such is the multiplicity of signifiers in maths. I really like the follow-up: ‘… we should also be looking for opportunities to shift the relationship that students have with mathematical objects… ‘ more. I think this is embodied in your first task, where you are shifting from thoughts about rectangles and area, to factors and combinatorics.
Have you thought about other examples of shared signifiers that are like square root (instruction and… label?)? + springs to mind… Or that are shared in some other way? And how could we teach these different signs without just pointing and telling, do you think?
I am currently interested in the power 2.. we call it ‘squared’ (a geometric representation) of course it doesn’t have to be, it can just be an numerical operation… This is talked about by Goold in MT17, and of course by David Pimm. I wonder what might be the consequences of these types of metaphors?
Well, thanks for the article, I enjoyed it, please let me know when you do the next one 🙂
Hi Sam and all
Yeah Mike some examples of what students do would add greatly to this and like you I am wondering about how Sam decides who gets which question and why he decides that, but I AM happy with tasks having some kind of lurking mathematical idea that Sam would like students to get a glimpse of. And a big question for me is how a teacher coordinates all the ideas and approaches in the class so that everyone gets some glimpse of that idea.
Hurrah and hurrah again for wanting numbers to be seen as products.
Tell me something interesting about 47, and 48, and 49 ….
And what would be the equivalent ‘trick’ with addition to the double and half trick for multiplication?
Have a good year Sam!
Thanks for the comments Anne.
The how I decide who gets what question and why is so important to teaching mathematics, so I wish I had a nice answer. Hopefully this example will give some insight into that;
With year 7’s the other my question to the class was “what pairs of numbers sum to 10?”, as I spoke to students and observed their workings a particular student had written down all the whole number pairs for this, my question was then “how many pairs are there?” to which she answered “11 or 6 if they are the same (referring to 8 + 2 as 2 + 8)”, at this stage it was clear to me that the student was happy that they had all the possible solutions. As we had recently studied negative numbers I wrote on her work 17 + ____ = 10. The student comfortably answered this, so I asked the “how many pairs” question again. Comparing this to another student in the same lesson, that when I spoke to and observed his work had already split his workings into three parts whole, decimals and negatives. My question of “how many pairs?” was answered with infinity, so my question was “how can we represent all these solutions?” He first spent some time writing this algebraically, with success, so I asked if a graph could show this.
I do make an effort to get to know my students (how successful this is I don’t know), but it with this in mind that I wish we kept our students together with the same teacher for their secondary education.
I’ll finish this on something I can answer confidently, 47 is odd and has an even number of factors, 48 is even and has an even number of factors, and 49 is odd and has an odd number of factors.
Thanks again,
Sam
Thanks for the comments Danny. There is another signifier that keeps me up at night, that is -, negative, subtract, minus! We read 4 – 6 = -2 as “four subtract six equals negative two” but why not “four subtract six equals subtract two”, the issues around negative don’t get any better, x^-1 = 1/x, but f^-1(x) is not 1/f(x), I think students first encounter this with sin, cos and tan.
I think there is a good argument to say we should just teach squaring as numerical operation. I think when teaching powers squaring is a terrible example to use anyway. 2 + 2 = 2 x 2 = 2^2 instead I favour what does 57^24 mean. It is also easy for a student to miss “three squared is three times three” in discussion, compared with “eighteen to the power of twenty seven is eighteen times eighteen times eighteen times … eighteen times”. What are your thoughts about power of 2.
Thanks again,
Sam
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